
The original integral ()
features \(n(n-1)/2\)  \(d\) -dimensional loop integrations and we have \((n-1)\)  delta functions to help us, with one remaining behind to enforce overall momentum conservation. This leaves the equivalent of \((n-1)(n-2)d/2\)  scalar integrals to perform.
In contrast, the parametrisations we derive feature fewer integrals: only \(n(n-1)/2\)  scalar parametric integrals,
one for each edge of the simplex.

By inverting the graph polynomials that arise,
we construct novel weight-shifting operators connecting solutions of the conformal Ward identities in spacetime dimension \(d\)  to new solutions in dimension \(d+2\) . Remarkably, these operators have a determinantal structure
based on the Cayley-Menger matrix familiar from
distance geometry.
In contrast, the well-known weight-shifting operators
introduced in [1]} preserve the spacetime dimension.
Operators mapping general correlators from \(d\rightarrow d+2\)  are, we believe, known only for the 3-point function where their existence can be seen from the triple-\(K\)  representation [2]}.
The new \(d\rightarrow d+2\)  operators we find thus generalise these results from 3-point to arbitrary \(n\) -point correlators.

